To check whether \(P_0(\mathbf {x}) \overset{d}{=}P_1(\mathbf {x})\)  (Condition 1 of Figure REF /Setting 2), standard hypothesis tests for distributional equality such as Kolmogorov-Smirnov [1]} can be used for each covariate. Non-parametric distributional distances (e.g., 2-Wasserstein, MMD [2]}) could help quantify to what extent \(P_0(\mathbf {x}) \overset{d}{=}P_1(\mathbf {x})\)  holds.
For distinguishing the high-risk group in particular, beyond hypothesis testing, incorporating domain knowledge on health disparities in the relevant covariates may be necessary.
To determine if the high-risk group is undertested (and therefore, performance gaps may arise), one may estimate \(\tau _a\)  (Condition 2, Figure REF ) via threshold tests [3]}, [4]}, [5]}.
For mitigation, the condition \(P_0(\mathbf {x}) \overset{d}{=}P_1(\mathbf {x})\)  is reminiscent of covariate shift in domain adaptation. Hence, standard approaches (e.g., reweighing methods [6]}, optimal transport [7]}, feature augmentation [8]}) may apply.
